1 Answer
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Taamei Haminhagim (citing Machazeh Avraham, by R. Avraham of Buczacz) says that it is because the ushpiza for the seventh day of Sukkos is King David, and he used to stay up all night studying Torah - so we do the same to evoke his corresponding Divine attribute.

R. Chaim Vital (Pri Eitz Chayim and Shaar Hakavanos, cited in Nitei Gavriel) give a different reason: because it is the last night before the final "sealing" of our judgment for the year.

IMHO, it would help to add to the second answer (based on HaRav Hayim Wital) that, because "it is the last night before the final 'sealing' [...]", we stay up all night specifically to study Torah or perform other mitzwot in an effort to increase our merits before "the final 'sealing' of our judgment".
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LeeOct 14 '14 at 17:19